Reflection, its law and Application

Light

It is an agent in which produces in us the sensation of sight. It is a form of energy. 

     No doubt this definition of light holds good only for a small portion of light, we shall be mainly concerned with light having this property. 

     Some terms connected with light. Some of the following terms shall be more oftenly during the study of optics. 

     1. Source. A body which emits light in all directions is said to be the source of light. The source may be a point one or an extended one. A source is of two kinds :

     (a) Self luminous. Self luminous source is the source which possesses light of its own, e.g., sun, electric arc, candle, etc. 

     (b) Non-luminous. It is a source of light which does not possess light of its own but receives light from an external source and scatters it to the surroundings, i.e., moon, page of your book, table, etc. 

      2. Medium. Substance through which light propagates or tends to propagate is called a medium. It is of following three kinds :

      (a) Transparent. It is a medium through which light can be propagated easily, e.g., glass, water, etc. 

     (b) Translucent. It is a medium through which light is propagated partially, e.g., paper, ground glass, etc. 

     (c) Opaque. It is a medium through which light cannot be propagated, e.g., wood, iron, etc. 

     3. Ray. The straight line path along which the light travels in a homogenous medium is called a ray. It is represented by an arrow head on a straight line, the arrow head giving the direction of propagation of light. 

     4. Beam. A number of rays combined together is called a beam. 

     5. Pencil. A narrow beam of rays is called a pencil. A pencil is of three kinds. 

     (a) Convergent. It is a pencil in which width of the pencil goes on decreasing as the rays proceed forward. 

     (b) Divergent. It is a pencil in which all the rays meet at a point when produced backward and the width of pencil goes on increasing as the rays proceed forward. 

     (c) Parallel. It is a pencil in which all the rays move parallel to each other and the width of the pencil remains constant throughout. 

Reflection

It is the property of light by virtue of which, light is sent after being obstructed by back into the same medium from which it is coming a surface. 

Fig. 1. Reflection at a plane surface. 

     Consider a ray SO, incident on a shining surface XY. The ray gets reflected along OI (Fig. 1). We define following terms in connection with the phenomenon of reflection. 

1. Incident ray (SO). Ray which approaches the shining surface. 

2. Reflected ray (IO). Ray which departs away from the shining surface. 

3. Normal (ON). It is a perpendicular drawn to the shining surface at the point of incidence O. 

4. Reflecting surface (XY). The shining surface which sends the ray back into the same medium is called reflecting surface. If nearly 100% of light is sent back the reflecting surface is said to be a mirror. 

5. Angle of incidence (i). It is the angle between the incident ray 'SO' and the normal 'ON' to the surface. 

6. Angle of reflection (r). It is the angle between the reflected ray OI and the normal ON. 

     

      The intensity of the reflected ray depends upon the nature, state of polish and smoothness of the reflecting surface. 

7. Angle of deviation during reflection = π - (i + r). It is the angle between initial incident ray after reflection and the reflected ray. 

Law of Reflection :

Phenomenon of reflection is governed by two laws called laws of reflection, given below :

     1. The incident ray, the reflected ray and the normal to the reflecting surface at the point of incidence, all lies in one plane and that plane is perpendicular to the reflecting surface. 

     2. The angle of incidence is equal to the angle of reflection, i.e., 

                        ㄥi = ㄥr. 

Relevant Reference

     We can see a piece of paper but we cannot see our image in the piece of paper, while we can see our image in a mirror. 


     The reason is that reflection occurring in the paper is diffused reflection because it's surface is not smooth. 

     If a parallel beam of light gets reflected from a rough surface (like paper) the reflected beam neither remains parallel nor has the same orientation as it was having before reflection. Thus, reflected beam is not able to reconstruct the image of the object. We just see general illumination. 

Fig. 2. Diffused reflection. 

Image formed by a plane mirror

     (a) Image of a point source. Consider a point source 'S' situated at a certain distance from the reflecting surface (mirror) XY. A ray of light SA incident normally (∴ i = 0) on XY is reflected back along AS (∴  r = 0) (Fig. 3). Another ray SO incident on XY obliquely at an angle of incidence i is reflected along OI, making an angle of reflection r, such that angle i and angle r are equal. Reflected rays AS and OI, meet at I' only when they are produced back. So I' is the virtual image of S as seen through the mirror XY. 

Fig. 3. Image of a point source. 

  

           ㄥNOS = ㄥOSA = ㄥi (alternate angles) 

           ㄥNOI = ㄥSI'O = ㄥr (corresponding angles) 

     According to the law of reflection

   

                ㄥi = ㄥr       ∴    ㄥOSA = ㄥAI'O

     In ∆ ASO and ∆ AI'O, OA is common. 

             ㄥOSA = ㄥAI'O      (just proved) 

             ㄥSAO = ㄥI'AO = 90°

     Therefore, the triangles are congruent

    

     ∴           SA = AI'

     i.e., the image I' lies as much behind the mirror as the object is in front of it. 

     Therefore, the image formed by a plane mirror has the following characteristics :

   

     (i) It is virtual. 

    (ii) It lies as much behind the mirror as the object lies in front of it. 

   (iii) It is of same size. 

   (iv) It is literally inverted. 

     Since the image is virtual it cannot be taken on a screen or photographed. 

     (b) Image of an extended source. Consider an extended source AB held before a plane mirror XY (Fig. 4). A beam of light from A is reflected into eye by the mirror when the beam meets the mirror at 'a' and 'a' '. Reflected rays meet at A₁ only when produced back. Thus, A₁ is the virtual image of A through the mirror. It may be noted that A and A₁, are both equidistant from the mirror. Similarly a beam of light starting from B, gets reflected from b, b' and enters the eye thereby giving a virtual image of B at B₁, when produced back. B and B₁ are also equidistant from the mirror. Thus, A₁B₁ is the virtual image of an extended object. 

Fig. 4. Image of an extended source. 

     The image as seen through a plane mirror is always laterally inverted, i.e., left side of the object appears as the right of the image and vice versa. If you raise your right hand in front of a mirror, the image will appears to raise the left hand. 

Rotation of a mirror

Consider a plane mirror in position X₁Y₁. An incident ray SO gets reflected along OI₁. Let the mirror rotates through an angle ∝ to the position  X₂Y₂. The reflected ray now goes to OI₂. Draw N₁O and N₂O perpendiculars to the two positions of the mirror (Fig. 5).

     ∴      ㄥN₁ON₂ = ㄥX₂OX₁ = ∝

     Applyinɡ the law of reflection in position X₂Y₂.

                  ㄥi₂ = ㄥr₂

     From the diaɡram 

                          ㄥi₂ = ㄥi₁ + ㄥ∝

and                   ㄥi₂ = ㄥr₁ − ㄥ∝ + ㄥβ

     ∴     ㄥi₁ + ㄥ∝ = ㄥr₁ − ㄥ∝ + ㄥβ

     But              ㄥi₁ = ㄥr₁     (law of reflection)

     Therefore,  ㄥ∝ = - ㄥ∝ + ㄥβ

or                        ㄥβ = 2ㄥ∝

Thus, the angle turned by the reflected ray is twice the angle turned by the mirror. 

Fig. 5. Rotation of a mirror. 

     So if a mirror turns through a certain angle the reflected ray turns through double that angle. 

Important notes In case of reflection from a plane mirror (i) The image is virtual and erect.  (ii) The image lies as much behind the mirror as the object is in front of it.  (iii) Magnification is 1. (iv) There is lateral inversion.  (v) If the mirror turns through certain angle, the reflected ray turns through double that angle.

Relevant Reference

1. Whenever a wave get reflected through a danser medium, an additional phase difference of π radian (or path difference of λ/2) is introduced

Fig. 6(i). 

yi = A sin (ωt - kx) 

yr = A sin [ωt - k (x + λ/2)] 

   = A sin [ωt - kx - kλ/2]

yr = A sin [ωt - kx - π ]         [∵ k = 2π/λ]

or    yr = A sin (ωt + kx)

2. [Refer Fig. 6(i)] if there are two mirrors inclined to each other at an angle θ. Then, number of images formed 'n', of an object are given as :

     (i) If 360/θ = even integer, n = (360/θ -1) ; for any position of object (A or B). 

    (ii) If 360/θ = odd integer, then :

         (a) n = (360/θ - 1); for object on the angle bisector of mirrors (A) 

         (b) n = 360/θ; for object anywhere else (B). 

Fig. 6(ii).

3. [Refer Fig. 6(iii)] If there are two mirrors perpendicular to each other, and a ray is incident on one of them such that, it suffers only one reflection from each of them, then the final ray will be antiparallel to the incident ray, no matter, what is the angle of incidence. 

Fig. 6(iii).

Conceptual questions with answers

Q. 1. What is the difference between regular and irregular reflection ? 

 

  Ans. The reflection is said to be the regular if it takes place in accordance with the law of reflection. It takes place from a smooth surface whether plane or spherical. When a beam of parallel rays is incident on an irregular surface, light is sent back into same medium but not in accordance with the law of reflection. This reflection is called irregular; or diffused surface. 

Q. 2. A ray of light is incident on a plane mirror at a certain point. The mirror is capable of rotation about an axis passing through some other point. Prove that if the mirror turns through a certain angle, the reflected ray turns through double the angle ? 

  Ans. Let XY be the position of mirror initially [Fig. 7]. A ray AO reflected at O goes along OB. 

Fig. 7.

                   ㄥAON = ㄥ BON

     ∴          ㄥAOY = ㄥBOX = ∝ (say) 

     Let the mirror be rotated to a new position XY' . The incident ray, now, gets reflected from O' along O'B'. O'N' is the new position of normal. 

     According to the law of reflection, 

                   ㄥAON = ㄥBON

     ∴           ㄥAOY = ㄥBOX = ∝

     Also      ㄥXO'O = ㄥAO'Y' = ∝

     ∴         ㄥB'O'O = 2∝'     

     Due to rotation of mirror the reflected ray turns through an angle BCB' = ፀ. 

                ㄥBCB' = ㄥO'CO = ፀ 

                ㄥO'OC = 180° - 2∝

     In ∆OO'C,     ፀ + 180° - 2∝ + 2∝' = 180°

or                                                    ፀ = 2(∝ -  ∝') 

     In ∆XOO',                                  ∝ = ф + ∝'

     ∴                                                ф = ∝ - ∝'

     Substituting for ф in (i)        ф = 2ф

Q. 3. A mirror was rotated through a certain angle but it was observed that the reflected ray did not turn through any angle. How would you explain ? 

  Ans. If the mirror rotates in its own place about the normal as the axis, there is no turning of reflected ray due to a rotation of mirror. 

Q. 4. Prove that if a ray of light is obliquely incident on one of the two mirrors inclined at angle 'ፀ' with each other, the net deviation of ray, after reflection from both the mirrors is independent of the angle of incidence ? 

  Ans. Consider a ray SA getting reflected from A at the first mirror and successively from B at the second mirror (Fig. 8). Finally, the ray proceeds towards BI. Let i₁ and i₂ be the angles of incidences at the two mirrors. 

           ㄥBAO = 90 - i₁  and  ㄥ ABO = 90 - i₂

           In ∆OAB, 

      

           θ + ㄥBAO + ㄥABO = 180°

           θ + 90 - i₁ + 90 - i₂ = 180°   

or       θ = i₁ + i₂

              Deviation at A = π - 2i₁

              Deviation at B = π - 2i₂

Fig. 8.

     Net deviation δ of the ray is given by 

            δ = π - 2i₁ + π - 2i₂ = 2π - 2(i₁ + i₂) 

or        δ = 2(π - θ) 

     Thus, the deviation only depends upon the angle between the two mirrors. It does not depends upon the angle of incidence.